Blow-up of solution of Cauchy problem for three-dimensional damped nonlinear hyperbolic equation

Asymptotic Behavior of Solutions to the Damped Nonlinear Hyperbolic Equation

“…Equation (1) fits the abstract framework of [1]. The local well-posedness for the Cauchy problem for (1), (2) in three-dimensional space was obtained by Chen and Da [2]. More precisely, they proved local existence and uniqueness of weak solutions to (1), (2) under the assumption that 0 ∈ 6 (R 3 ), 1 ∈ 4 (R 3 ).…”

“…The local well-posedness for the Cauchy problem for (1), (2) in three-dimensional space was obtained by Chen and Da [2]. More precisely, they proved local existence and uniqueness of weak solutions to (1), (2) under the assumption that 0 ∈ 6 (R 3 ), 1 ∈ 4 (R 3 ). Local existence and uniqueness of classical solutions to (1), (2) were also established, provided that 0 ∈ 12 (R 3 ), 1 ∈ 10 (R 3 ).…”

“…More precisely, they proved local existence and uniqueness of weak solutions to (1), (2) under the assumption that 0 ∈ 6 (R 3 ), 1 ∈ 4 (R 3 ). Local existence and uniqueness of classical solutions to (1), (2) were also established, provided that 0 ∈ 12 (R 3 ), 1 ∈ 10 (R 3 ). Their method is to first establish local-intime well-posedness of a periodic version of (1), (2) and then construc a solution to (1), (2) as a limit of periodic solutions with divergent periods.…”

“…Local existence and uniqueness of classical solutions to (1), (2) were also established, provided that 0 ∈ 12 (R 3 ), 1 ∈ 10 (R 3 ). Their method is to first establish local-intime well-posedness of a periodic version of (1), (2) and then construc a solution to (1), (2) as a limit of periodic solutions with divergent periods. This paper also arrived at some sufficient conditions for blow-up of the solution in finite time, and an example was given.…”

Local existence and global nonexistence theorems for a damped nonlinear hyperbolic equation

“…The existence and uniqueness of the Cauchy problem for Eq. (1.4) are proved in [6]. The initial boundary value problem (1.1)-(1.3) describes the vibration of a 3-dimensional nonlinear damped wave with fixed boundary.…”